Late stage droplet growth - McGill Physicsgrant/Papers/PhysicaA204_770.pdf · Late stage droplet growth Jian Hua Yao, K.R. Elder, Hong Guo and Martin Grant Centre for the Physics

Physica A 204 (1994) 770-788

North-Holland

SSDI 037%4371(93)E0403-2

Late stage droplet growth

Jian Hua Yao, K.R. Elder, Hong Guo and Martin Grant

Centre for the Physics of Materials, Physics Department, Rutherford Building, McGill University, 3600 rue University, Montrtal, Quebec, Canada H3A 2T8

The phenomenon of Ostwald ripening, where large droplets in a supersaturated solution

grow at the expense of small droplets, was theoretically explained by Lifshitz and Slyozov.

Modern theories provide extensions of this classic work to the situation where the volume

fraction of the phase appearing as droplets is appreciable. This has been done by perturbation

expansions, mean-field theory, and numerical methods. Herein, those recent developments

are reviewed.

1. Introduction

Nucleation and growth is the most commonplace of first-order phase

transitions. It occurs when a binary mixture is cooled rapidly from a disordered

phase into a two-phase coexistence region, if the volume fraction 4 of the

minority component is small. It is useful to think of this process in terms of

early and late time regimes. Initially, the dynamics of the phase transition is

controlled by the nucleation of droplets from a supercooled or supersaturated

solution, and little growth of those droplets occurs as the supersaturation is

relieved by nucleation. For late times, no signification nucleation takes place,

instead the phase transition involves the competitive growth of many droplets,

so as to minimize surface energy. There is no sharply defined time when these

two regimes switch, instead for intermediate times nucleation takes place at the

same time as droplet growth. Here we focus on droplet growth in the late

stages of the phase transition. Hence we consider a two-phase system where

one phase forms a background matrix, and the minority phase consists of many

droplets, as shown in fig. 1. As time t evolves, the total number of droplets

decreases and the average droplet radius R(t) increases: Large droplets grow

by the condensation of material diffused through the matrix from small

evaporating droplets. This phenomenon is called Ostwald ripening [l].

The growth process is clearly shown in fig. 1, where configurations from a

numerical simulation of Ostwald ripening are presented. Since the volume

fraction is constant, surface area is decreased by the evaporation of droplets

037%4371/94/$07.00 @ 1994 - Elsevier Science B.V. All rights reserved

J.H. Yao et al. I Late stage droplet growth 771

Id1

Fig. 1. Sketch of Ostwald ripening in two dimensions. Shaded circles represent the droplets (the

minority component) fixed in two-dimensional space. As time evolves (via the numerical method

discussed in section 3) from (a) to (d), the total number of droplets decreases and the average

droplet radius increases, but the volume fraction of droplets q5 (the shaded area) is constant.

smaller than an average size. The thermodynamic force conjugate to the surface area is the curvature l/R, which enters through the chemical potential [2]. This driving force leads particles to diffuse from regions of high curvature to those of low, thus reducing the total interfacial free energy of the two-phase system. These are the main features of Ostwald ripening.

The object of a theory of Ostwald ripening is to determine the dynamics and morphology of droplet growth. In what follows, we focus on the average droplet radius R(t) and the number density n(R, t), although other quantities, such as the structure factor, are also of interest. For many years after Ostwald’s original discovery in 1901 of the phenomenon [l], the understanding of growth was only qualitative, as described above. Early attempts [3,4] did not succeed because they did not properly address diffusion between droplets through the matrix. The major advance in the theory of Ostwald ripening was made by the

772 J.H. Yao et al. I Late stage droplet growth

classic works of Lifshitz and Slyozov [5] and Wagner [6] more than thirty years

ago.

Herein, we will review work done since that time, particularly our own

recent work. It is an honor to have this paper included in a collection

commemorating the research of Professor Kyozi Kawasaki. Professor

Kawasaki’s contributions to all aspects of the modern theory of the kinetics of

first-order phase transitions have been profound. In particular, with his

collaborators Tokuyama and Enomoto, he has given a firm grounding to the

modern theory of Ostwald ripening [7].

The physics of the problem is straightforward. After nucleation has ceased,

one has many small droplets of various sizes, where all concentrations are at

their local equilibrium values. That is, the droplets are at the concentration of

the minority phase, while far from all droplets, the matrix has the concen-

tration of the majority phase. Hence the only driving force for equilibration is

the surface free energy of all the droplets; the bulk free energies are already

equilibrated. The difficulty is that the total concentration of either phase is

conserved, i.e., the total volume of all the droplets is the product of the

minority phase’s volume fraction and the total volume of the system:

c $TRQ = const. = +L3, (1)

where i is an index running over all N(t) droplets, Ri is the radius of the ith

droplet, and L3 is the system volume. This means that, since all droplets are

spherical and have locally equilibrated their surface energy, the total system of

many droplets can only minimize surface energy further by some small droplets

getting smaller, and large droplets getting larger. This requires the diffusion of

material through the matrix between widely separated droplets. This long-

range process makes the problem technically challenging. Indeed, as we will

see, it becomes analogous to a time-dependent, multi-charge Coulomb prob-

lem.

Nevertheless one can still anticipate the main results, and hence focus on

issues which theory and experiment can usefully address. First, it is clear that

the interface of a droplet moves so as to minimize its surface area. The

thermodynamic force conjugate to the area is interface curvature l/R, so we

expect that the velocity u = dRldt m l/R. But the relief of this thermodynamic

force, by redistributing matter through the matrix to other droplets, requires

diffusion through that matrix, which is limited by l/R. Hence we anticipate

u - 1/R2. Indeed, this gives the correct growth law from dimensional analysis,

R(t) = (Kt)1’3, (2)


for the average droplet size, where the constant K is called the coarsening rate. Now since R + ~0 as t- CQ, all quantities that depend on length should scale with this diverging length. For instance, let the number density of droplets be

n(R, t) = x 6(R - Ri(t)) > (3)

so that

I dR n(R, t) = N(t) , (4)

the total number of droplets, which evidently decreases in time. Clearly the total number N(t) - [LIR(t)13 ’ m three dimensions, so that the scaling form for the number density is

n(R, t) = E-4g(RIl?) , (5)

where l?(t) is given above, and g(z) is the scaling function. The arguments giving these results are rather simple, and therefore turn out

to be general. It is expected that thermal fluctuations are irrelevant in first- order transitions (they affect small length scales such as the thermal correlation length, which do not diverge), so that one finds the same scaling and form for algebraic growth for a large class of systems where a conservation law limits phase separation. In that sense, these results are superuniversal, since they are valid for different dimensions of space, and bridge the nominal universality classes of critical dynamics [8]. Of course, by this it is only meant that scaling and growth involve power laws determined by dimensional analysis, without anomalous dimensions.

Hence, the task of theory - besides the fundamental justification of algebraic growth of the average radius, and the scaling of the number density-is to calculate the numerical value of the coarsening rate K, the form of the scaling function g(z), and other observable quantities. We shall be concerned with the dependence of these quantities on the dimension of space, and in particular, their dependence on volume fraction 4. It is straightforward to anticipate the effect of an appreciable volume fraction: since there is more minority-phase material present, droplets will grow faster, and droplets of increasingly different relative sizes will be present. That is, K should increase while g(z) broadens with 4. To obtain quantitative results, we must begin with the fundamental equations of motion.


2. Theoretical approaches

The starting point of the dynamic theory is to describe the diffusion of

material between droplets.

excess concentration field c

s=V’c-source=O,

The diffusion equation for the (dimensionless) #I

1s

(6)

where the source is due to all the droplets, and we shall consider the

steady-state limit where the concentration is everywhere relaxed to its local

equilibrium value. Each droplet serves as a source or sink of diffusion current,

so we may rewrite this equation as

v2c = 47r c Q,(t) S(r - r,) ) (7)

where r, points to the center of each droplet, and the currents Qi at the surface

of each droplet will be determined below. We have chosen this notation to

promote the analogy between this problem and a Coulomb problem, which is

striking when one notes that the conserved concentration obeys

$ dr c(r) = 0 ,

so that one must have

2 Q;=O

Henceforth we will sometimes call the currents Qi charges.

The boundary condition at the surface of each droplet is

1 4.) = R,

(8)

(9)

(10)

for Ir - ril = Ri, which is called the Gibbs-Thomson condition, while far from

all droplets the concentration field is its value in the majority phase.

‘I Units of length and time arc given in terms of a characteristic length 1 -(D - l)yV,iR,T (where D is the dimension of space, y is the surface tension, V, is the molar volume, Rg is the gas

constant, and T is temperature) and a characteristic time 12/9C,Vm (where 9 is the diffusion

constant and C, is the concentration above a flat interface). The dimensionless excess concen-

tration field is c(r) = [C(r) - C,]lC,, where C is the concentration.


Using these boundary conditions, it is worthwhile to write down the formal

solution of the diffusion equation in the steady-state limit as

N Q- (11)

where Q,(t) is an integration constant determined by the conservation of

charge (it is of course the concentration in the majority phase). At the surface

of each droplet, the Gibbs-Thomson condition gives

f=QO-~-i=.&~. I I I I

(12)

Therefore, once one knows the sizes and positions of all the droplets, it is

straightforward to calculate the charges Qi of each droplet. Note also that,

coincidentally, the excess concentration at the surface of the droplet, and the

contribution from diffusion from other droplets, both vary as l/R, when the

droplets are widely separated, i.e., in the limit as the volume fraction vanishes.

The rate of change of the volume of each droplet is simply determined by

those charges, which give the diffusion current JD over that droplet, i.e.,

(13)

where the first equality over the surface Si of droplet i has been changed to a

volume integral in the second equality. On using the form for the source

above, we obtain

(14)

Formally, the problem is now complete; after determining the charges of

each droplet from a set of initial conditions, one can use the equation of

motion to find how each droplet evolves with time. Indeed, as shown originally

by Voorhees and Glicksman [9], whose convenient notation we are following

here, one can simply solve the equations numerically, as we will discuss below.

However, most approaches consider a mean-field theory. Indeed, it is from this

starting point that the theory of Lifshitz and Slyozov begins, although many

approximations have already been made. For example, note that we have

neglected all fluctuations. The important fluctuation not included is nucleation,

since we have assumed that rate is now negligible. While this is a good

description of many experimental systems with small volume fractions, as the


volume fraction becomes large, it should be noted that nucleation can

conceivably compete with droplet growth. It should also be noted that we have

neglected the possibilities of fluid convection in a liquid matrix or stress

relaxation if the droplets are solid crystallites. Nevertheless, the formulation of

Ostwald ripening is clear and nontrivial in this form; generalizations can be

made to incorporate other effects if they are found to be important experimen-

tally.

We will now briefly review the mean-field theory of Lifshitz and Slyozov.

Consider a droplet separated an arbitrary distance from all other droplets, i.e.,

consider the limit in which the volume fraction 4 -+ 0. Then in eq. (12) above

lrj -T~I+~, and

+Qo_g. I

(15)

Rewriting this as Q, = QoRi - 1, and summing over all droplets gives 0 =

Q, Ci Ri - Ci, or

1 Q0=R(t)7 (16)

where the bar denotes an average. Then the equation of motion for Ri is

(17)

Note that a droplet smaller than the average size shrinks and disappears, while

one larger than that size will continue to grow.

To make further progress, we consider the number density n(R, t), where

continuity in R-space implies

an@, 4 --+&(n(R,t)~)=O. at

It is straightforward to solve for the growth law R(t) and the number density

n(R, t) if one makes the scaling ansatz as above of

n(R, t) = Re4g(RIR), (19)

where g(z) is the scaling function, and all the time dependence of n enters

through R(t). Using the equation of motion above, the continuity equation for


the number density becomes separable in the two independent variables t and

z = R/R(t):

dR R2(t) dt =

(2 - z)g(z) + (z’ - z) (dg/dz)

4z3g(z) + z4 dg/dz

Hence

dR R2(t) 7 = K , (21)

where K is a constant independent of t or z, which is determined by positivity

and normalization of g. The classic results of Lifshitz and Slyozov for C#J += 0 in

dimension D = 3 are then recovered:

R(t) = (Kt)“3 , (22)

for late times, where K = 4/9, and

g(z) = (34e/2s’3)z2exp[-l/(l -fz)]/[(z +3)7’3(: -z)~~‘~] if O<z<+ , o

otherwise .

(23)

These features, power-law growth and scaling, are now considered universal

characteristics of the kinetics of a first-order phase transition [8], and have

been observed experimentally [ 10-121.

Nevertheless, it has proved difficult to rigorously test the Lifshitz-Slyozov

theory by experiment or by numerical simulation. Experiments typically study

volume fractions which are appreciably larger than zero, while numerical work

has the additional problem of being practically difficult in three dimensions.

Furthermore, large scale numerical work has been limited by previous

computer facilities. Hence modern theories give the form of the coarsening

rate K = K(4), and the scaling function g = g(z, 4). For the most part, analytic

extensions have been based either on ad hoc assumptions [13-151, or on

perturbative expansions in 4, typically taken to order fi [7,16]. Numerical

work of good accuracy has become feasible only recently.

The phenomenological theory of Ardell [13] provides a physical though ad

hoc method to extend the Lifshitz-Slyozov result to nonzero volume fractions.

When the volume fraction is appreciable, Ardell noted that the droplets of

different “charges”, that is with currents acting as sources and sinks, will

screen each other, just as electric charges do. Say the average distance between

droplets, which is determined by the volume fraction, is 5. Ardell suggested


that the diffusion current from other droplets was screened effectively at a

distance 5/2 from the surface of a droplet. This then replaces one of the

boundary conditions on the concentration field.

Indeed, this idea of screening appears in all modern approaches, and

Ardell’s ensuing theory is physically appealing. However, it is not rigorous; for

example, as noted by him, the choice of 512, rather than some other fraction of

5, is arbitrary. Nevertheless, the major features of Ostwald ripening with

nonzero volume fraction, the increase in K and broadening of g(z) with +, are

captured, although the dependence is not quantitatively correct, as we shall

see.

A rigorous treatment by perturbation in the volume fraction has been done

mainly by two groups, Marqusee and Ross (MR) [16], and Tokuyama,

Kawasaki and Enonmoto (TKE) [7]. In the MR approach, one perturbs in the

bare Coulomb propagator from the Laplace equation using a formal scattering

expansion, while TKE make use of a modified growth law and separately

perturb the so-called drift and soft-collision terms, respectively. Both theories

are technically demanding, and give results to leading order in fi, and so

apply to exceedingly small volume fractions. There are small but we suspect

unobservable differences between the predictions of these theories in the

extremely small volume fraction limit to which they apply. These are due to the

difference between the way the two groups treat direct correlations between

droplets (called soft collisions by TKE).

In somewhat the same spirit, the possible effect of two-particle correlations

was studied by Mardar in an ambitious theory [17]. While his starting point

may prove to be useful to future work, many of his approximations are difficult

to justify, and his results are not consistent with the rigorous perturbation

theories of MR and TKE.

Recently, there has been interest in two-dimensional systems, where ex-

perimental work can be done, for example, on droplets absorbed on a surface

or films. The issues in D = 2 are similar to those above, but there are

additional difficulties. Theoretically, the technical problem is the logarithmic

divergence encountered in the propagator of the Laplace equation. Indeed, the

original LSW approach cannot be applied to two-dimensional systems, since

the general solution for two-dimensional steady-state diffusion in eq. (6),

C(Y) - In Y, cannot simultaneously satisfy the boundary conditions at the surface

of a droplet and far from all droplets. One must identify the appropriate length

to cut off the range of diffusion far from all droplets. Various theories have

been proposed.

In the zero-volume fraction limit, Rogers and Desai (RD) [18] proposed a

non-steady-state approach, where that extra length is essentially the evolving

droplet size. (We shall continue to call 4 a volume fraction although it is an


area fraction in D = 2.) Solving the time-dependent diffusion equation they obtained the modified growth law for the average droplet size, R - (tlln t)li3 for 4 = 0.

For nonzero volume fractions, evidently the appropriate length to make the propagator well defined is due to the nonzero volume fraction. Theories have been proposed by Ardell [13], Marqusee [ 191, and Zheng and Gunton [20]. None of these approaches are rigorous, in the sense of the perturbation theories mentioned above. They all obtain a scaling form for the number density with the conventional R - t”3 result. Ardell has generalized his three dimensional phenomenological theory to D = 2. The approach taken by Marqusee in two dimensions [19] is significantly different from his three- dimensional work. The idea used in this work is similar to that of Ardell in that a screening length is introduced in the diffusion equation, whose value is then self-consistently determined. Finally, to study two-droplet correlations in two dimensions, Zheng and Gunton [20] combined Marqusee’s two-dimensional theory with a generalization of Mardar’s theory.

We now review our work, which applies to D = 2 or D = 3 [21-241. The results of that theory cannot be distinguished from the perturbation theories in the very small volume fraction limit, and since our approach is by mean field, we have results to all orders in the volume fraction. Our main approximation is the complete neglect of direct correlations between droplets. Following the next section, we will compare our work to the results of other theories and numerical work, finding that it fares rather well. One other advantage of our approach is that it is simple analytically, and hence straightforward to generalize. In fact, we have generalized it in other works to consider excitonic systems [23] and systems where phase separation is limited by surfactants [24].

For nonzero volume fractions, we approximate the spatial correlations contained in eq. (12) above by mean-field theory. We make use of the fact that, for non-zero +, the steady-state problem resembles a homogeneous electron gas, since droplets interact via the Laplace equation in the steady-state limit and charge neutrality is invoked. We introduce screening effects among the droplets and approximate the many-droplet correlation effects in the same manner as the Thomas-Fermi mechanism for Coulomb systems. We replace the sum over multiple droplets with an effective potential $, which is determined self-consistently. That is, we find the excess concentration at the surface of a droplet to be

where 5 is a screening length due to the other droplets, related to the volume


fraction of the minority phase, where $( ,.$+ m)+ l/R,. It only remains to

establish the form of $, and the conditions determining 5 by the volume

fraction. As one would expect, the effective potential turns out to have the

Yukawa form I,!I = (l/R)[exp(-R/t)] in three dimensions.

To obtain an effective potential, we reconsider the growth law. First we

replace the spatial coupling between droplets with an effective current J(R, R') which only depends on the size of all droplets. Then

~~R:=C J(R,,R,)AC(Rj). i

(25)

where Ac(Rj) is the individual driving force due to each droplet given by the

concentrations deviation from its value in the majority phase (i.e., AC(R) =

Q,, - c(R)). W’th 1 in our mean-field approximation, all droplets are equivalent,

so J(R,R') = J(R)6,,,,, and

where it remains to determine J(R,). The mean-field approximation results

from the assumption that the flux determining the growth rate for each droplet

is only proportional to the difference between the boundary concentration and

the average bulk concentration.

We now postulate an equation of motion for the local concentration field

C(T, t) in the vicinity of the ith droplet. The most simple form it can satisfy is

~=V’C+‘C+S-~~Q~~(~-T~). (27)

Herein, local diffusion is modified by the effective diffusion field from the

other droplets, giving rise to the screening length 5 and the background field

St2. These quantities can be related to J(R) by integrating the equation above

and comparing it with eq. (26), i.e.,

5

2

J(R)+,f)dR, (28)

(29)


Consistency between these two equations is enforced by

for the effective field. Now it is straightforward to obtain an effective field I,!J which gives the effect of the other charges Qj away from the droplet. Furthermore, from the growth law, this will determine the coupling current J, which turns out to be proportional to the inverse of $.

Eqs. (26) and (27) completely specify our mean-field approximation; indeed, they are the only approximations needed to solve the equations in the steady-state limit. Their form implies that we consider a one-body problem without correlations. A systematic derivation of these equations from first principles would be valuable, since corrections to our equations, in- volving correlations, could be calculated. However, we have not been able to obtain such a derivation, although, of course, a coarse graining of the microscopic equations, with the requirement that only a one-body distribution function is involved, will lead to our self-consistent starting point.

The analysis follows the same route as the above for Lifshitz and Slyozov, with the potential derived from the solution of eq. (27) for c at the surface of the droplet being of the Yukawa form in three dimensions, as noted above. One finds that the equations are again separable, and the growth law and number density are straightforwardly determined. The complete details, including analysis for two-dimensional systems, are given in our original papers [21-241.

As in other theories, we find that K increases and g(z) is broader as the volume fraction increases. For example, in two dimensions, as 4 -+ 0, we find R(t) - [tlln(+--“2)]*‘3. Hence the (ln t) singularity found in the non-steady- state calculation of Rogers and Desai for 4 = 0 is replaced by a (ln 4) piece in our steady-state approach. Presumably one could derive a crossover form for very small 4 where the Rogers and Desai result is obeyed for early times, crossing over to this result for later times. Finally, we note that solutions can only be found for our mean-field equations for restricted values of volume fraction: Our approach is self-consistently correct for 4 < 0.06 in three dimensions and 4 s 0.085 in two dimensions.

To conclude this section, we note that all theories agree on the qualitative behavior for nonzero-volume fractions, but there are large quantitative differences. After briefly reviewing numerical approaches in the next section, we will compare theories to numerical work and experiment .


3. Numerical simulations

For small volume fractions, a beautiful numerical scheme based on a

potential method was introduced in 1984 by Voorhees and Glicksman [9]. They

carried out numerical simulations by an Ewald-sum technique. Unfortunately

that work, and subsequent numerical work [25], was hampered by the

computing facilities available at that time, so that statistics were unreliable.

However, recently we [21,22] have been able to carry out larger scale

simulations in both two and three dimensions, in which 1000 droplets were

considered. Some of these results will be reported below. We should note that

other methods are possible, for example Chakrabarti et al. have integrated the

nonlinear Langevin equation corresponding to the universality class of Ostwald

ripening, and obtained interesting results [26]. Nevertheless, at present the

potential method is the most accurate to our knowledge.

The basic equations for the three-dimensional simulation are given above.

Given an initial configuration of droplets, one inverts the matrix of eq. (12),

which formally solves the diffusion problem, to obtain the charges {Qi} from

the droplet radii {Ri} at positions {ri}. (The extra current Q, is determined by

conservation of charge.) Then the sizes of droplets are updated by the equation

of motion eq. (14).

However, simply integrating the equations in this form is awkward numeri-

cally because of long-range interactions. For example, in two dimensions, one

has

g=Q,+Q,lnR,+ i Q,lnIr,-rjI. I i=l.i#1

(31)

A tractable form of this equation can be obtained by means of Ewald-sum

techniques [10,21,22], which give

$= Q, + Qj(ln(Rj/L) + i ’ PJ-r’z d,.‘_ i$d,.‘)

0 1

x

_ i Qj

r=l,i#j

1 $&.f -5: Qi z. ep~~4L* eik.(r~-rI) ,

lr,-r,llL

(32)

where L* is the system size. The remainder of the numerical analysis is

straightforward though tedious. For example, in order to keep the volume


fraction constant during the iterations, the time increment must be determined self-consistently by shrinking, at most, one droplet in each single time step.

4. Comparison of different theories

Fig. 2 gives the time evolution of the average droplet radius obtained from our simulations, showing that the numerical data is consistent with R(t) =

[K(4) v3. The values for the coarsening rate obtained by the fits to fig. 2 are given as a function of 4 in fig. 3, in which the numerical values for K are compared with various theoretical predictions. One can see that the theories of TKE, MR, and Marqusee well explain the data at small volume fractions, while our theory continues to work well at somewhat larger volume fractions.

Comparisons of theoretical predictions for the scaled distribution function g(z) in D = 3 with simulations and an experimental measurement [12] are presented in fig. 4, and in D = 2 in fig. 5. Different symbols correspond to distribution functions at different times as noted in the caption. Scaling is confirmed by the universal curve to which the data from different times collapse. Again, one can see that the theories of TKE, MR, and Marqusee well explain the data at small volume fractions, while our theory continues to work well at somewhat larger volume fractions. The good fit of Ardell’s theory in d = 2 is accidental, since the coarsening rate is not well described by that theory.

It should be noted that the main difference between our theory and the perturbation theories is the inclusion of terms to all orders in 4 in our mean-field theory. The perturbation theories and ours give essentially the same result to leading order in the expansion in j/$. However, our numerical data indicates that higher-order volume fraction effects, beyond this order, are important for volume fractions as low as 1% .

5. Conclusions

In summary, modern approaches to the theory of Ostwald ripening have advanced to the point where one now has a quantitative understanding of this phenomena. The theories reviewed here can be tested experimentally by many methods. In particular, this would allow the study of the importance of effects which have been neglected here. It is impossible to say with certainty what those effects might be. Perhaps correlations will be of importance, but we

J.H. Yao et al. I Late stage droplet growth

“0 13 26 39 52

t

8.0

0.0 0.0 6.0 12.0

lo3 t Fig. 2. Results of our numerical simulations for the time evolution of average droplet size (I?‘(f) - R’(O)) vs. t for (a) Q = 0, 0.01 and 0.05 in D = 3; (b) 4 = 0.01, 0.05 and 0.10 in D = 2. The straight lines indicate that the time evolution of the average droplet radius is consistent with

I?(t) = [I?‘(O) + K(4) t]li3, allowing us to estimate the coarsening rate K(4).

rather suspect it is our starting point itself which will have to be modified. For

example, in the nucleation of a solid from a liquid, we expect stress and strain

effects to be important, which have been neglected from our starting point.

s z

J.H. YQO et al. J Late stage dropret growth

3.8 ““,““,““,“”

1.2

0.8

0.4

I I I I _./ _....- .-- D=2 (b)

_.A

,./ *./

*./

_..* ..a*

. ..* /.‘.

_/

. ..’ . ..- . ..* /.- / /*

,’ /‘- . ..* a l

0 II / . ..*- /ye ___---- _______-----

I’i; . ._____:__------ __/----

0.0 + 0.0 I.7 3.4 5.1 6.8 8.5x1o-2

0

785

Fig. 3. Plots of coarsening rates K vs. q5 in (a) D = 3, and (b) D = 2. In (a) (K(4)/K(O) vs. c$), the

dotted, dashed, long-dashed, dash-dotted, and solid lines correspond respectively to results of MR

[16], TKE [7]#‘, Ardell [13], Mardar [17], and us, denoted YEGG. The lines for MR and TKE are almost superimposed. The symbols correspond to our simulation results. In (b) (K(4) vs. c$), dotted, dashed, long-dashed, and solid lines correspond respectively to results of Ardell [13],

Marqusee [19], ZG [20], and us, denoted YEGG. The symbols represent our simulation results.

** Here, we only present TKE’s mean-field results, called the drift-term approximation by them.


2.0

= 1.0 GJ

0.0

z=Rli? Fig. 4. Comparisons of the distribution functions g(z) between different theories, our simulations,

and an experiment is displayed in D = 3 for 4 = 0.01 in (a); for 0.05 in (b). The symbols (except

the solid circles) are simulation results. The different symbols correspond to different times when

the number of remaining droplets N =600, 500, 400 and 300. The dotted, long-dashed, dash-

dotted, and solid lines are the respective predictions of MR [16], TKE [7], Ardell [13], Mardar

[17], and us, denoted YEGG ([21,22] in D = 3), where the lines for MR and TKE superimpose. In

(b), the solid circles are the experimental distribution function at very late times [12] for 4 - 0.05.


1.0

0.0

z=R/R Fig. 5. Plot of the scaled normalized distribution function g(z) vs. the scaled radius z = K/R

for D = 2. Dotted, dashed, and solid lines correspond respectively to Marqusee [19], Ardell

[13], and us, denoted YEGG [21,22]. Circles, squares, and triangles give the scaled normalized

distribution functions from the simulation, corresponding respectively to different times when

the number of remaining droplets N=500, 400, and 300, for 4 = 0.01 in (a); for 4 = 0.05 in (b).


Acknowledgements

We thank Mohamed Laradji for collaboration on some of the work reviewed

here. This work was supported by the Natural Sciences and Engineering

Research Council of Canada and le Fonds pour la Formation de Chercheurs et

1’Aide B la Recherche de la Province de QuCbec.

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Late stage droplet growth - McGill Physicsgrant/Papers/PhysicaA204_770.pdf · Late stage droplet growth Jian Hua Yao, K.R. Elder, Hong Guo and Martin Grant Centre for the Physics

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